3step scattering consider an incoming wave from the left scattering off the step" potential of the form viso iso v() = v. i> 0 for v > 0.
(a) find the reflection coefficient for e (b) find the reflection coefficient for e> vo.
(c) for such a potential, the transmission coefficient does not have the simple form f /a because the potential does not go to zero as 2 0 . as a result, the transmitted wave travels at a different speed. show that in this case t v do this by noting that is the probability of finding the wave in relative to some incident probability in < 0. for a plane wave, we can define an incident probability in a finite region-for example, in a particular aren over a time interval dt. use this to argue that t = where ur, uy are the velocities of the transmitted and incident waves, and then use your knowledge of the velocity of a plane wave in terms of its energy.
(d) show t = in a more clever way, by using an equivalent formulation of t in terms of probability currents, t = where j. and j are the probability currents of the transmitted and incident waves, respectively. probability current is useful for once in our lives.
(e) find t for e (f) find t explicitly for e > v, and verify that t +r=1.